(This is a work in progess…) Nth roots of Unity \(n^{th}\) roots of unity is given by the equation:\(\large (1)^{1\over n}=e^{i\frac{2k\pi}{n}}\), \(\large k=0,1,\dots,n-1\) Applet for visualizing nth roots of unity Nth Cyclotomic Polynomial \(n^{th}\) cyclotomic polynomial, for any positive integer \(n\), is the unique irreducible polynomial with integer coefficients that is a divisor of \(x^{n}-1\) and is not a divisor of \(x^{k}-1\) for any \(k < n\). Its roots are…

(This is a work in progess…) Rearrangement Theorem: \(R\bf{G} = \bf{G}\)  \((\forall R \in \bf{G})\). Equivalent statement about the columns being \(\bf{G} R = \bf{G}\)  \((\forall R \in \bf{G})\) Each row and each column in the group multiplication table lists each of the group elements once and only once. From this, it follows that no two elements may be in the identical location in two rows or two columns. Thus,…